Space-time Equation Research Articles

Single wave solutions of the fractional Landau-Ginzburg-Higgs equation in space-time with accuracy via the beta derivative and mEDAM approach

<p>The nonlinear wave behavior in the tropical and mid-latitude troposphere has been simulated using the space-time fractional Landau-Ginzburg-Higgs model. These waves are the consequence of interactions between equatorial and mid-latitude waves, fluid flow in dynamic systems, weak scattering, and extended linkages. The mEDAM method has been used to obtain new and extended closed-form solitary wave solutions of the previously published nonlinear fractional partial differential equation via the beta derivative. A wave transformation converts the fractional-order equation into an ordinary differential equation. Several soliton, single, kink, double, triple, anti-kink, and other soliton types are examples of known conventional wave shapes. The answers are displayed using the latest Python code, which enhances the usage of 2D and 3D plotlines, as well as contour plotlines, to emphasise the tangible utility of the solutions. The results of the study are clear, flexible, and easier to replicate.</p>

One-dimensional Carrollian fluids. Part I. Carroll-Galilei duality

Galilean and Carrollian algebras acting on two-dimensional Newton-Cartan and Carrollian manifolds are isomorphic. A consequence of this property is a duality correspondence between one-dimensional Galilean and Carrollian fluids. We describe the dynamics of these systems as they emerge from the relevant limits of Lorentzian hydrodynamics, and explore the advertised duality relationship. This interchanges longitudinal and transverse directions with respect to the flow velocity, and permutes equilibrium and out-of-equilibrium observables, unveiling specific features of Carrollian physics. We investigate the action of local hydrodynamic-frame transformations in the Galilean and Carrollian configurations, i.e. dual Galilean and Carrollian local boosts, and comment on their potential breaking. Emphasis is laid on the additional geometric elements that are necessary to attain complete systems of hydrodynamic equations in Newton-Cartan and Carroll spacetimes. Our analysis is conducted in general Cartan frames as well as in more explicit coordinates, specifically suited to Galilean or Carrollian use.

Optimal approximations for the free boundary problems of the space-time fractional Black-Scholes equations using a combined physics-informed neural network

The combined physics-informed neural network is employed to deal with the free boundary problems of fractional Black-Scholes equations. The solution assumption and the loss function are determined, the transfer learning is borrowed, the combined neural network with data enhancement layer is designed, then the classical Black-Scholes model is numerically solved and the comparative analysis of numerical results under different neural networks is made. For further insight into the long-term memory of fluctuation, the free boundary problems of the space-time Black-Scholes equations under Caputo, Caputo-Fabrizio and Atangana-Baleanu-Caputo fractional derivatives are studied. The corresponding empirical analyses are presented and the optimal exercise boundaries of American put option are simulated. The market analysis shows that introducing fractional calculus tools and neural network algorithms into American put option pricing can yield more realistic prediction results. The work provides a viable method for subsequent researchers to study American option pricing using fractional calculus and neural networks combined with true market data and to deal with the free boundary problems in other research fields.

Evolution of fermion resonance in thick brane

Abstract In this work, we investigate the numerical evolution of massive Kaluza–Klein (KK) modes of a Dirac field on a thick brane. We deduce the Dirac equation in five-dimensional spacetime, and obtain the time-dependent evolution equation and Schrödinger-like equation of the extra-dimensional component. We use the Dirac KK resonances as the initial data and study the corresponding dynamics. By monitoring the decay law of the left- and right-chiral KK resonances, we compute the corresponding lifetimes and find that there could exist long-lived KK modes on the brane. Especially, for the lightest KK resonance with a large coupling parameter and a large three momentum, it will have an extremely long lifetime.

Global Existence in the Critical Space for the Space-Time Monopole Equations

We prove the global existence of solutions in the critical space \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1({\\mathbb {R}})$$\\end{document} for space-time monopole equations under the Lorenz gauge condition. The existence of a global solution was showed in our previous paper. Here we adapt a different function space to allow for a simplified proof. We first prove the local existence in \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1({\\mathbb {R}})$$\\end{document} and extend the local solution to the global one.

Towards field theory of multiple D0-branes. Hamiltonian mechanics and quantization of simplest 3D prototype of multiple D0-brane system

Recently we have constructed a completely supersymmetric nonlinear action possessing the properties expected from multiple D0-brane system. Its quantization should result in an interesting supersymmetric field theory in the (super)space with additional matrix coordinates which can provide an important insights in the study of String Theory. As a first stage toward this aim, in this paper we construct the Hamiltonian mechanics and perform covariant quantization of the simplest three dimensional counterpart of the ten dimensional multiple D0-brane model. We obtain a supersymmetric system of equations in a (super)spacetime enlarged by bosonic and fermionic matrix coordinates which appears as a result of such quantization and discuss some of its properties.

Global existence and blowup of smooth solutions to the semilinear wave equations in FLRW spacetime

We are interested in the semilinear wave equations evolving in the expanding spacetimes with Friedmann–Lemaître–Robertson–Walker (FLRW) metric. By the weighted energy estimate, we show that when the nonlinearity depends on the time derivative of the unknown, the equation admits a global smooth solution if the spacetime is undergoing accelerated expansion. While the solution will blowup in the sense of some averaged quantity if the expanding rate is not fast enough. When the nonlinearity depends on the space derivatives of the unknown or the unknown itself, we can show that the solution will blowup in finite time even though the expanding rate is fast enough (accelerated expansion). Our results show that the semilinear wave equations in FLRW spacetimes have different properties from the famous Glassey and Strauss conjectures in flat or asymptotically flat spacetimes.

Toward local Madelung mechanics in spacetime

It has recently been shown that relativistic quantum theory leads to a local interpretation of quantum mechanics wherein the universal wavefunction in configuration space is entirely replaced with an ensemble of local fluid equations in spacetime. For want of a fully relativistic quantum fluid treatment, we develop a model using the nonrelativistic Madelung equations, and obtain conditions for them to be local in spacetime. Every particle in the Madelung fluid is equally real, and has a definite position, momentum, kinetic energy, and potential energy. These are obtained by defining quantum momentum and kinetic energy densities for the fluid and separating the momentum into average and symmetric parts, and kinetic energy into classical kinetic and quantum potential parts. The two types of momentum naturally give rise to a single classical kinetic energy density, which contains the expected kinetic energy, even for stationary states, and we define the reduced quantum potential as the remaining part of the quantum kinetic energy density. We treat the quantum potential as a novel mode of internal energy storage within the fluid particles, which explains most of the nonclassical behavior of the Madelung fluid. For example, we show that in tunneling phenomena, the quantum potential negates the barrier so that nothing prevents the fluid from flowing through. We show how energy flows and transforms in this model, and that enabling local conservation of energy requires defining a quantum potential energy current that flows through the fluid rather than only flowing with it. The nonrelativistic treatment generally contains singularities in the velocity field, which undermines the goal of local dynamics, but we expect a proper relativistic treatment will bound the fluid particle velocities at c.

Principle of minimal singularity for Green’s functions

Analytic continuations of integer-valued parameters can lead to profound insights, such as angular momentum in Regge theory, the number of replicas in spin glasses, the number of internal degrees of freedom, the spacetime dimension in dimensional regularization, and Wilson’s renormalization group. In this work, we consider a new kind of analytic continuation of correlation functions, inspired by two recent approaches to underdetermined Dyson-Schwinger equations in D-dimensional spacetime. If the Green’s functions Gn=⟨ϕn⟩ admit analytic continuation to complex values of n, the two different approaches are unified by a novel principle for self-consistent problems: Singularities in the complex plane should be minimal. This principle manifests as the merging of different branches of Green’s functions in the quartic theories. For D=0, we obtain the closed-form solutions of the general gϕm theories, including the cases with complex coupling constant g or noninteger power m. For D=1, we derive rapidly convergent results for the Hermitian quartic and non-Hermitian cubic theories by minimizing the complexity of the singularity at n=∞. Published by the American Physical Society 2024

The nonlinear wave dynamics of the space-time fractional van der Waals equation via three analytical methods

In this paper, we explore the new exact soliton solutions of the truncated M-fractional nonlinear (1 + 1)-dimensional van der Waals equation by applying the expa function method, extended (G′/G)-expansion method, and modified simplest equation method. The concerned equation is a challenging problem in the modeling of molecules and materials. Noncovalent van der Waals or dispersion forces are frequent and have an impact on the structure, dynamics, stability, and function of molecules and materials in biology, chemistry, materials science, and physics. The results obtained are verified and represented by two-dimensional, three-dimensional, and contour graphs. These results are newer than the existing results in the literature due to the use of fractional derivative. The achieved solutions will be of high significance in the interaction of quantum-mechanical fluctuations, granular matter, and other areas of van der Waals equation applications. Therefore, the obtained solutions are valuable for future studies of this model.

Soliton solutions and a bi-Hamiltonian structure of the fifth-order nonlocal reverse-spacetime Sasa-Satsuma-type hierarchy via the Riemann-Hilbert approach

<p>Our objective is to explore the intricacies of a nonlinear nonlocal fifth-order scalar Sasa-Satsuma equation in reverse spacetime which is rooted in a nonlocal $ 5 \times 5 $ matrix AKNS spectral problem. Starting with this spectral problem, we derive both local and nonlocal symmetry relations through rotations within a defined group. We then formulate a specific type of Riemann-Hilbert problem, facilitating the generation of soliton solutions. These solutions are generated by utilizing vectors that reside in the kernel of the matrix Jost solutions. Under the condition where reflection coefficients are null, the jump matrix reduces to the identity, leading to soliton solutions via the corresponding Riemann-Hilbert problem. The explicit formulas of these soliton solutions enable a comprehensive exploration of their dynamics.</p>

On the Dirac-like equation in 7-component space-time and generalized Clifford-Dirac algebra

The generalized Dirac equation related to 7-component space-time with one time coordinate and six space coordinates has been introduced. Three 8-component Dirac equations have been derived from the same 256-dimensional Clifford-Dirac matrix algebra. Corresponding Clifford-Dirac algebra is considered in the Pauli-Dirac representation of $8 \times 8$ gamma matrices. It is proved that this matrix algebra over the field of real numbers has 256-dimensional basis and it is isomorphic to geometric $\textit{C}\ell^{\texttt{R}}$(1,7) algebra. The corresponding gamma matrix representation of 45-dimensional $\mathrm{SO}(1,9)$ algebra is derived and the way of its generalization to the $\mathrm{SO}(m,n)$ algebra is demonstrated. The Klein-Gordon equation in 7-component space-time is considered as well. The way of corresponding consideration of the Maxwell equations and of equations for an arbitrary spin is indicated.

On the consequences of Raychaudhuri equation in Kantowski-Sachs space-time

The paper aims to study the geometry and physics of the Raychaudhuri equation (RE) in the background of a homogeneous and anisotropic space–time described by Kantowski–Sachs (KS) metric. Role of anisotropy/shear in the context of convergence and possible avoidance of singularity has been analyzed subject to a physically motivated constraint. Moreover, using a suitable transformation the first order RE has been converted to a second order differential equation analogous to a Harmonic Oscillator and criterion for convergence has been shown to be associated with the time varying frequency of the Oscillator. Finally, the paper points out a geometric and physical notion of anisotropy along with their corresponding behavior towards convergence.

An $$L_q(L_p)$$-theory for space-time non-local equations generated by Lévy processes with low intensity of small jumps

An $$L_q(L_p)$$-theory for space-time non-local equations generated by Lévy processes with low intensity of small jumps

Some space-time fractional bright–dark solitons and propagation manipulations for a fractional Gross–Pitaevskii equation with an external potential

Some nonautonomous bright–dark solitons (NBDSs) and nonautonomous controllable behaviors in the conformable space-time fractional Gross–Pitaevskii (FGP) equation with some external potentials are derived. We consider the relations between the space-time FGP equation and the fractional nonlinear Schrödinger equation and analyze the properties of the obtained equation with group velocity dispersion and spatiotemporal dispersion. Then, some constraint conditions of the valid soliton solutions are given. Furthermore, we consider the effect of α and β in NBDSs of the space-time FGP equation. Some fractional spatial–temporal controlling wave prolong phenomena are considered, and some different propagation dynamics are generated via the different parameters α and β. We study novel shape bright soliton solution, novel ‘h’-shape dark soliton and some interactions of nonautonomous bright–dark solitons. The reported results of some novel interactions are considered, which can explain some models of the electrical and optical fields.

Shock waves in (1 + 1-dimensional) curved space-time

ABSTRACT Shock jump conditions are widely used to solve various astrophysical problems. From the hydrodynamic equation, we derive the jump condition and the Taub adiabat equation in curve space-time for both time-like and space-like shocks. We find that the change in entropy for the weak shocks for curved space-time is small, similar to that for flat space-time. We also find that for general relativistic space-like shocks, the Chapman–Jouguet point does not necessarily correspond to the sonic point for downstream matter, unlike the special relativistic case. To analyse the shock wave solution for the curved space-time, one needs the information of metric potentials describing the space-time, which is assumed to be a neutron star for the present work. Assuming a shock wave is generated at the star’s centre, and as it propagates outward, it combusts nuclear matter to quark matter. We find that the general relativistic treatment of shock conditions is necessary to study shocks in neutron stars so that the results are consistent. We also find that with such general relativistic treatment, the combustion process in neutron stars is always a detonation.

Light propagation in Kerr spacetime

We explicitly solve the equations for the propagation of an electromagnetic wave up to the subleading order geometric optics expansion in the Kerr spacetime. This is done in two nontrivial steps. We first construct a set of parallel propagated null tetrad in Kerr spacetime. Two of the components of such tetrad give the propagation and polarization of an electromagnetic wave in geometric optics approximation. Then we use the parallel propagated tetrad to solve the modified trajectory equation in Kerr spacetime. We obtain the wavelength-dependent deviation of the trajectory of electromagnetic waves, which gives the mathematical description of the gravitational spin Hall effect in Kerr spacetime.

Partially coherent sources whose coherent modes are spatiotemporal optical vortex beams

We analyze three non-stationary partially coherent sources whose coherent modes are spatiotemporal optical vortex (STOV) beams. Using spatiotemporal (ST) Bessel–Gauss and Laguerre–Gauss beams (STOV-carrying solutions to the space-time paraxial wave equation) as eigenfunctions in the coherent-modes representation of the mutual coherence function, we derive the ST versions of J 0-Bessel-correlated, I n -Bessel-correlated, and twisted Gaussian Schell-model beams. We model, in simulation, these ST random beams via their coherent-modes expansions, compare and contrast the simulated results to theory, and analyze/discuss their free-space propagation characteristics. The work presented in this paper will be useful for simulating or physically generating these ST beams for use in applications or future studies.

Evolution of scalar field resonances in a braneworld

In this work, we investigate the numerical evolution of massive Kaluza–Klein (KK) modes of a scalar field in a thick brane. We derive the Klein–Gordon equation in five-dimensional spacetime, and obtain the evolution equation and the Schrödinger-like equation. With the resonances of the scalar KK modes as the initial data, the scalar field is evolved with the maximally dissipative boundary condition. The results show that there are scalar KK resonant particles with long life on the brane, which indicates that these resonances might be regarded as a candidate for dark matter.

HKLL for the non-normalizable mode

We discuss various aspects of HKLL bulk reconstruction for the free scalar field in AdSd+1. First, we consider the spacelike reconstruction kernel for the non-normalizable mode in global coordinates. We construct it as a mode sum. In even bulk dimensions, this can be reproduced using a chordal Green’s function approach that we propose. This puts the global AdS results for the non-normalizable mode on an equal footing with results in the literature for the normalizable mode. In Poincaré AdS, we present explicit mode sum results in general even and odd dimensions for both normalizable and non-normalizable kernels. For generic scaling dimension ∆, these can be re-written in a form that matches with the global AdS results via an antipodal mapping, plus a remainder. We are not aware of a general argument in the literature for dropping these remainder terms, but we note that a slight complexification of a boundary spatial coordinate (which we call an iϵ prescription) allows us to do so in cases where ∆ is (half-) integer. Since the non-normalizable mode turns on a source in the CFT, our primary motivation for considering it is as a step towards understanding linear wave equations in general spacetimes from a holographic perspective. But when the scaling dimension ∆ is in the Breitenlohner-Freedman window, we note that the construction has some interesting features within AdS/CFT.